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3 years ago

The minute hand of a clock is 3 inches long. How far does the tip of the minute hand move in 40 minutes?

Mathematics

1 answer:

Mkey [24]3 years ago

6 0

Answer:

4π in ≈ 12.57 in

Step-by-step explanation:

One full revolution (2π radians) is made in 60 minutes. In 40 minutes, the hand moves through an angle θ of (40/60)(2π) = 4π/3 radians. The length of the arc is ...

s = rθ = (3 in)(4π/3) = 4π in ≈ 12.57 in

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Consider the integral Integral from 0 to 1 e Superscript 6 x Baseline dx with nequals 25 . a. Find the trapezoid rule approximat

photoshop1234 [79]

Answer:

With n = 25, $\int_{0}^{1}e^{6 x}\ dx \approx 67.3930999748549$

With n = 50, $\int_{0}^{1}e^{6 x}\ dx \approx 67.1519320308594$

b. $\int_{0}^{1}e^{6 x}\ dx \approx 67.0715427161943$

The absolute error in the trapezoid rule is 0.08047

The absolute error in the Simpson's rule is 0.00008

Step-by-step explanation:

a. To approximate the integral $\int_{0}^{1}e^{6 x}\ dx$ using n = 25 with the trapezoid rule you must:

The trapezoidal rule states that

$\int_{a}^{b}f(x)dx\approx\frac{\Delta{x}}{2}\left(f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)\right)$

where $\Delta{x}=\frac{b-a}{n}$

We have that a = 0, b = 1, n = 25.

Therefore,

$\Delta{x}=\frac{1-0}{25}=\frac{1}{25}$

We need to divide the interval [0,1] into n = 25 sub-intervals of length $\Delta{x}=\frac{1}{25}$ , with the following endpoints:

$a=0, \frac{1}{25}, \frac{2}{25},...,\frac{23}{25}, \frac{24}{25}, 1=b$

Now, we just evaluate the function at these endpoints:

$f\left(x_{0}\right)=f(a)=f\left(0\right)=1=1$

$2f\left(x_{1}\right)=2f\left(\frac{1}{25}\right)=2 e^{\frac{6}{25}}=2.54249830064281$

$2f\left(x_{2}\right)=2f\left(\frac{2}{25}\right)=2 e^{\frac{12}{25}}=3.23214880438579$

...

$2f\left(x_{24}\right)=2f\left(\frac{24}{25}\right)=2 e^{\frac{144}{25}}=634.696657835701$

$f\left(x_{25}\right)=f(b)=f\left(1\right)=e^{6}=403.428793492735$

Applying the trapezoid rule formula we get

$\int_{0}^{1}e^{6 x}\ dx \approx \frac{1}{50}(1+2.54249830064281+3.23214880438579+...+634.696657835701+403.428793492735)\approx 67.3930999748549$

To approximate the integral $\int_{0}^{1}e^{6 x}\ dx$ using n = 50 with the trapezoid rule you must:

We have that a = 0, b = 1, n = 50.

Therefore,

$\Delta{x}=\frac{1-0}{50}=\frac{1}{50}$

We need to divide the interval [0,1] into n = 50 sub-intervals of length $\Delta{x}=\frac{1}{50}$ , with the following endpoints:

$a=0, \frac{1}{50}, \frac{1}{25},...,\frac{24}{25}, \frac{49}{50}, 1=b$

Now, we just evaluate the function at these endpoints:

$f\left(x_{0}\right)=f(a)=f\left(0\right)=1=1$

$2f\left(x_{1}\right)=2f\left(\frac{1}{50}\right)=2 e^{\frac{3}{25}}=2.25499370315875$

$2f\left(x_{2}\right)=2f\left(\frac{1}{25}\right)=2 e^{\frac{6}{25}}=2.54249830064281$

...

$2f\left(x_{49}\right)=2f\left(\frac{49}{50}\right)=2 e^{\frac{147}{25}}=715.618483417705$

$f\left(x_{50}\right)=f(b)=f\left(1\right)=e^{6}=403.428793492735$

Applying the trapezoid rule formula we get

$\int_{0}^{1}e^{6 x}\ dx \approx \frac{1}{100}(1+2.25499370315875+2.54249830064281+...+715.618483417705+403.428793492735) \approx 67.1519320308594$

b. To approximate the integral $\int_{0}^{1}e^{6 x}\ dx$ using 2n with the Simpson's rule you must:

The Simpson's rule states that

$\int_{a}^{b}f(x)dx\approx \\\frac{\Delta{x}}{3}\left(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right)$

where $\Delta{x}=\frac{b-a}{n}$

We have that a = 0, b = 1, n = 50

Therefore,

$\Delta{x}=\frac{1-0}{50}=\frac{1}{50}$

We need to divide the interval [0,1] into n = 50 sub-intervals of length $\Delta{x}=\frac{1}{50}$ , with the following endpoints:

$a=0, \frac{1}{50}, \frac{1}{25},...,\frac{24}{25}, \frac{49}{50}, 1=b$

Now, we just evaluate the function at these endpoints:

$f\left(x_{0}\right)=f(a)=f\left(0\right)=1=1$

$4f\left(x_{1}\right)=4f\left(\frac{1}{50}\right)=4 e^{\frac{3}{25}}=4.5099874063175$

$2f\left(x_{2}\right)=2f\left(\frac{1}{25}\right)=2 e^{\frac{6}{25}}=2.54249830064281$

...

$4f\left(x_{49}\right)=4f\left(\frac{49}{50}\right)=4 e^{\frac{147}{25}}=1431.23696683541$

$f\left(x_{50}\right)=f(b)=f\left(1\right)=e^{6}=403.428793492735$

Applying the Simpson's rule formula we get

$\int_{0}^{1}e^{6 x}\ dx \approx \frac{1}{150}(1+4.5099874063175+2.54249830064281+...+1431.23696683541+403.428793492735) \approx 67.0715427161943$

c. If B is our estimate of some quantity having an actual value of A, then the absolute error is given by $|A-B|$

The absolute error in the trapezoid rule is

The calculated value is

$\int _0^1e^{6\:x}\:dx=\frac{e^6-1}{6} \approx 67.0714655821225$

and our estimate is 67.1519320308594

Thus, the absolute error is given by

$|67.0714655821225-67.1519320308594|=0.08047$

The absolute error in the Simpson's rule is

$|67.0714655821225-67.0715427161943|=0.00008$

6 0

3 years ago

I need the answer now please

xxMikexx [17]

Answer:

180/147

Step-by-step explanation:

We need to find the additive inverse of the given numbers.

For finding it we may simply out -1 as multiplication in front of the terms .

SOLUTION 1 :-

→ 3/-7 - 11/21

→ -3/7 -11/21

→ 3×-3 - 11 /21

→ -9-11/21

→ -20/21

SOLUTION 2 :-

→ 9/5 ÷ 7/5

→ 9/5 × 5/7

→ 9/7

→ Product of nos . = -20/21 × -9/7

→ Ans = 180/147

7 0

3 years ago

Dado dos ángulos complementarios, uno mide (2x + 10) y el otro (x + 20),

rusak2 [61]

Answer:

El valor de x es igual a 20 o x = 20.

Step-by-step explanation:

Lo primero que se debe saber es que dos ángulos complementarios suman un ángulo recto o 90º.

Supongamos que el valor de un ángulo $\\ \alpha$ y un ángulo $\\ \beta$ valen:

$\\ \alpha = 2x + 10$ [1]

$\\ \beta = x + 20$ [2]

Como la suma de $\\ \alpha + \beta = 90$ [3]

Entonces

$\\ \alpha + \beta = (2x + 10) + (x + 20) = 90$

Sumamos los factores comunes entre si:

$\\ (2x + x) + (10 + 20) = 90$

Para la primera expresión debemos recordar que se suman sólo los coeficientes. Así:

$\\ (2 + 1)x + (10 + 20) = 90$

$\\ 3x + 30 = 90$

Para despejar la incógnita x, debemos tener en cuenta que una igualdad no se altera si se suma, se resta, se multiplica o divide un mismo valor a cada lado de ella. Por esta razón, para despejar 3x, lo primero que podemos hacer es sumar -30 a cada lado de la expresión (lo que es igual a restar 30 a cada lado de la misma). Así tenemos:

$\\ 3x + 30 - 30 = 90 - 30$

$\\ 3x + 0 = 90 - 30$

$\\ 3x = 60$

Ahora dividimos cada miembro de la igualdad entre 3 (o multiplicamos cada lado de la igualdad por $\\ \frac{1}{3}$ ):

$\\ \frac{3}{3}x = \frac{60}{3}$

Como sabemos que:

$\\ \frac{3}{3} = 1$

Entonces:

$\\ 1*x = \frac{60}{3}$

$\\ x = \frac{60}{3}$

$\\ x = 20$

De esta manera, el valor de x es igual a 20 o x = 20.

Lo anterior lo podemos comprobar considerando las ecuaciones [1], [2] y [3]. Así tenemos que:

$\\ \alpha = 2x + 10$ [1]

Sustituimos x por el valor de 20:

$\\ \alpha = 2*20 + 10 = 40 + 10 = 50$

$\\ \beta = x + 20$ [2]

Hacemos lo mismo para [2]:

$\\ \beta = 20 + 20$

$\\ \beta = 40$

De esta manera:

$\\ \alpha + \beta = 90$ [3]

$\\ 50 + 40 = 90$

4 0

3 years ago

Help me pls I really really need help with this I'm so bad at math

MAVERICK [17]

Question 7: Answer J (7/8 - 1/2 = 3/8)

6 0

3 years ago

A restaurant uses 3 pounds of pasta for a party of 14 people. If they are cooking pasta for a party of 56 people, what should th

azamat

If you would like to know how many pounds of pasta the restaurant will need, you can calculate this using the following steps:

3 pounds of pasta ... 14 people
x pounds of pasta = ? ... 56 people

3 * 56 = 14 * x /14
x = 3 * 56 / 14
x = 12 pounds of pasta

The correct result would be 12 pounds of pasta.

3 0

3 years ago

Read 2 more answers

The minute hand of a clock is 3 inches long. How far does the tip of the minute hand move in 40 minutes​?

The minute hand of a clock is 3 inches long. How far does the tip of the minute hand move in 40 minutes?